Dirichlet series
| L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s + 5·9-s − 4·10-s + 4·11-s − 6·13-s − 4·16-s + 6·17-s − 10·18-s + 4·20-s − 8·22-s − 12·23-s − 7·25-s + 12·26-s + 8·32-s − 12·34-s + 10·36-s + 6·37-s + 8·44-s + 10·45-s + 24·46-s + 5·49-s + 14·50-s − 12·52-s + 8·55-s + 20·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s + 5/3·9-s − 1.26·10-s + 1.20·11-s − 1.66·13-s − 16-s + 1.45·17-s − 2.35·18-s + 0.894·20-s − 1.70·22-s − 2.50·23-s − 7/5·25-s + 2.35·26-s + 1.41·32-s − 2.05·34-s + 5/3·36-s + 0.986·37-s + 1.20·44-s + 1.49·45-s + 3.53·46-s + 5/7·49-s + 1.97·50-s − 1.66·52-s + 1.07·55-s + 2.60·59-s + ⋯ |
Functional equation
Invariants
| Degree: | \(4\) |
| Conductor: | \(10816\) = \(2^{6} \cdot 13^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.689637\) |
| Root analytic conductor: | \(0.911287\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 10816,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6496674880\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6496674880\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 2 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) | |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) | |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) | |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) | |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) | |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) | |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) | |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) | |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) | |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) | |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) | |
| 71 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) | |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) | |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) | |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) | |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | |
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Imaginary part of the first few zeros on the critical line
−14.03138070170820377342404104863, −13.65256715401797432042833844266, −12.93652707664178297812045231949, −12.35409975707263402349255573602, −11.83154549957557831819446561218, −11.55168050213096256500942544377, −10.25760428819074643973910806183, −10.12002502784925675527084533184, −9.756467649995391558827101569606, −9.613508474892148105860213063569, −8.742668970694198924173312210402, −7.948701491373206449920631763674, −7.43213155263816955805373058027, −7.15030621521244837397206939435, −6.19223577176150942201911223395, −5.66492165023256605873583598016, −4.45607701323885764529889498638, −3.97335010471256464363848913478, −2.23083718496778732570938837681, −1.50989599007397932630514486544, 1.50989599007397932630514486544, 2.23083718496778732570938837681, 3.97335010471256464363848913478, 4.45607701323885764529889498638, 5.66492165023256605873583598016, 6.19223577176150942201911223395, 7.15030621521244837397206939435, 7.43213155263816955805373058027, 7.948701491373206449920631763674, 8.742668970694198924173312210402, 9.613508474892148105860213063569, 9.756467649995391558827101569606, 10.12002502784925675527084533184, 10.25760428819074643973910806183, 11.55168050213096256500942544377, 11.83154549957557831819446561218, 12.35409975707263402349255573602, 12.93652707664178297812045231949, 13.65256715401797432042833844266, 14.03138070170820377342404104863